arXiv subjects

Kenneth H. Karlsen

Publications and source records attributed to Kenneth H. Karlsen.

At least 19 recordsLinked to original sources

arXiv2026

Structure-preserving LDG methods for linear and nonlinear transport equations with gradient noise

We develop local discontinuous Galerkin (LDG) methods for conservation laws with heterogeneous stochastic fluxes, where the Stratonovich-driven transport terms may be linear or nonlinear. Such equations arise, for example, in simplified turbulence models, mean field games, and fluctuating hydrodynamics. Starting from the Itô formulation, we construct semi-discretizations that build the cancellation mechanism of transport noise into the numerical method. At the discrete energy level, the second-order Stratonovich-Itô correction is balanced by the quadratic variation, up to numerical flux terms, so that the hyperbolic stability structure is retained. Suitable numerical fluxes yield discrete energy conservation or energy dissipation, valid either pathwise or in expectation. The resulting high-order schemes are proved well posed through stability estimates combined with a Khasminskii-type argument, without imposing linear growth assumptions. Numerical experiments confirm stability and high-order accuracy.

math.NA↗

arXiv2026

Regularity of velocity averages in kinetic equations with heterogeneity

This study investigates the regularity of kinetic equations with spatial heterogeneity. Recent progress has shown that velocity averages of weak solutions $h$ in $L^p$ ($p>1$) are strongly $L^1_{\text{loc}}$ compact under the natural non-degeneracy condition. We establish regularity estimates for equations with an $\boldsymbol{x}$-dependent drift vector $\mathfrak{f} = \mathfrak{f}(\boldsymbol{x}, \boldsymbolλ)$, which satisfies a quantitative version of the non-degeneracy condition. We prove that $(t,\boldsymbol{x}) \mapsto \int ρ(\boldsymbolλ) h(t,\boldsymbol{x},\boldsymbolλ)\, d\boldsymbolλ$, for any sufficiently regular $ρ(\cdot)$, belongs to the fractional Sobolev space $W_{\text{loc}}^{β,r}$, for some regularity $β\in (0,1)$ and integrability $r \geq 1$ exponents. While such estimates have long been known for $\boldsymbol{x}$-independent drift vectors $\mathfrak{f}=\mathfrak{f}(\boldsymbolλ)$, this is the first quantitative regularity estimate in a general heterogeneous setting. As an application, we obtain a regularity estimate for entropy solutions to heterogeneous conservation laws with nonlinear flux and $L^\infty$ initial data.

math.AP↗

arXiv2026

Velocity averaging under minimal conditions for deterministic and stochastic kinetic equations with irregular drift

This study investigates the $L^1_{\operatorname{loc}}$ compactness of velocity averages of sequences of solutions $\{u_n\}$ for a class of kinetic equations. The equations are examined within both deterministic and stochastic heterogeneous environments. The primary objective is to deduce velocity averaging results under conditions on $u_n$ and the drift ${\mathfrak f}={\mathfrak f}(t,{\boldsymbol x},{\boldsymbol λ})$ that are more lenient than those stipulated in previous studies. The main outcome permits the inclusion of highly irregular drift vectors ${\mathfrak f} \in L^q$ that adhere to a general non-degeneracy condition. Moreover, the sequence $\{u_n\}$ is uniformly bounded in $L^p$ -- for an exponent $p$ allowed to be strictly smaller than $2$ -- under the requirement $\frac{1}{p} + \frac{1}{q} < 1$. Resolving the matter of strong compactness in velocity averages, considering these assumptions, has remained an open problem for a long time. The cornerstone of our work's progress lies in the strategic employment of the broader concept of $H$-distributions, moving beyond the traditional reliance on $H$-measures. Notably, our study represents one of the first significant uses of $H$-distributions in this context.

math.AP↗

arXiv2026

Semi-discrete heat equations with variable coefficients and the parametrix method

We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the classical continuous setting benefits from Gaussian estimates of the constant coefficient heat kernel, such estimates are not available in the semi-discrete context. To address this complication, we derive estimates involving products of heavy-tailed Lorentz (also known as Cauchy) probability densities. These Lorentzian estimates provide a sufficient handle on certain iterated convolutions involving Bessel functions, enabling us to achieve convergence of the parametrix approach.

math.NA↗

arXiv2025

Generic regularity and Lipschitz metric for a two-component Novikov system

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions between peakon-like structures. Our study has two main objectives: first, to analyze the generic regularity of global conservative solutions; and second, to construct a new metric that guarantees the Lipschitz continuity of the flow. Building on the geometric framework developed by Bressan and Chen for quasilinear second-order wave equations, we prove that the solution retains $C^k$ regularity away from a finite number of piecewise $C^{k-1}$ characteristic curves. Furthermore, we provide a description of the solution behavior in the vicinity of these curves. By introducing a Finsler norm on tangent vectors in the space of solutions, expressed in the transformed Bressan-Constantin variables, we introduce a Lipschitz metric representing the minimal energy transportation cost between two solutions.

math.AP↗

arXiv2025

A duality approach to the fractional Laplacian with measure data

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-Δ)^s v = μ\quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $μ$ is a bounded Radon measure whose support is compactly contained in $\mathbb{R}^N$, $N\geq2$, and $-(Δ)^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.

math.AP↗

arXiv2025

Global semigroup of conservative weak solutions of the two-component Novikov equation

We study the Cauchy problem for the two-component Novikov system with initial data $u_0, v_0$ in $H^1(\mathbb{R})$ such that the product $(\partial_x u_0)\partial_x v_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $(\partial_x u)^2dx$, $(\partial_x v)^2dx$, and $\left((\partial_x u)^2(\partial_x v)^2\right)dx$, which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm.

math.AP↗

arXiv2025

On the regularity of entropy solutions to stochastic degenerate parabolic equations

We study the regularity of entropy solutions for quasilinear parabolic equations with anisotropic degeneracy and stochastic forcing. Building on previous works, we establish space-time regularity under a non-degeneracy condition that does not require an assumption on the derivative of the symbol of the corresponding kinetic equation, a restriction imposed in earlier studies. This allows us to obtain regularity results for certain equations not accounted for by prior theory, albeit with reduced regularity exponents. Our approach uses a kinetic formulation with two transport equations, one of second order and one of first order, leveraging a form of "parabolic regularity" inherent in these equations that was not utilized in previous studies.

math.AP↗

arXiv2025

Convergent finite difference schemes for stochastic transport equations

We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The $L^2$ estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.

math.NA↗

arXiv2025

Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes

The Yamada-Watanabe theory provides a robust framework for understanding stochastic equations driven by Wiener processes. Despite its comprehensive treatment in the literature, the applicability of the theory to SPDEs driven by Poisson random measures or, more generally, Lévy processes remains significantly less explored, with only a handful of results addressing this context. In this work, we leverage a result by Kurtz to demonstrate that the existence of a martingale solution combined with pathwise uniqueness implies the existence of a unique strong solution for SPDEs driven by both a Wiener process and a Poisson random measure. Our discussion is set within the variational framework, where the SPDE under consideration may be nonlinear. This work is influenced by earlier research conducted by the second author alongside de Bouard and Ondreját.

math.PR↗

arXiv2024

Stochastic electromechanical bidomain model

We analyze a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearized elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearized elastic behavior and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo--Galerkin method. We utilize a stochastic adaptation of de Rham's theorem to deduce the weak convergence of the pressure approximations.

math.AP↗

arXiv2024

Convergence of stochastic integrals with applications to transport equations and conservation laws with noise

Convergence of stochastic integrals driven by Wiener processes $W_n$, with $W_n \to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is on the convergence of the form $\int_0^T V_n\, \mathrm{d} W_n \to \int_0^T V\, \mathrm{d} W$, where $\{V_n\}$ is bounded in $L^p(Ω\times [0,T];X)$ for a Banach space $X$ and some finite $p > 2$. This is challenging when $V_n$ converges to $V$ weakly in the temporal variable. We supply convergence results to handle stochastic integral limits when strong temporal convergence is lacking. A key tool is a uniform mean $L^1$ time translation estimate on $V_n$, an estimate that is easily verified in many SPDEs. However, this estimate alone does not guarantee strong compactness of $(ω,t)\mapsto V_n(ω,t)$. Our findings, especially pertinent to equations exhibiting singular behavior, are substantiated by establishing several stability results for stochastic transport equations and conservation laws.

math.PR↗

arXiv2024

On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $(m,n)\in C^{k}(\mathbb{R}) \cap W^{k,1}(\mathbb{R})$ with $k\in\mathbb{N}\cup\{0\}$.This system extends the celebrated Fokas-Olver-Rosenau-Qiao equation, and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $$ \partial_t m(t,x)= \partial_x[m(t,x)(u(t,x)-\partial_xu(t,x)) (u(-t,-x)+\partial_x(u(-t,-x)))], $$ where $m(t,x)=\left(1-\partial_{x}^2\right)u(t,x)$. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $C^k\cap W^{k,1}$. Moreover, we derive criteria for blow-up of the local solution in this class.

math.AP↗

arXiv2024

Well-posedness of stochastic continuity equations on Riemannian manifolds

We analyze continuity equations with Stratonovich stochasticity, $\partial ρ+ div_h \left[ ρ\circ\left(u(t,x)+\sum_{i=1}^N a_i(x) \dot W_i(t) \right) \right]=0$, defined on a smooth closed Riemannian manifold $M$ with metric $h$. The velocity field $u$ is perturbed by Gaussian noise terms $\dot W_1(t),\ldots,\dot W_N(t)$ driven by smooth spatially dependent vector fields $a_1(x),\ldots,a_N(x)$ on $M$. The velocity $u$ belongs to $L^1_t W^{1,2}_x$ with $div_h u$ bounded in $L^p_{t,x}$ for $p>d+2$, where $d$ is the dimension of $M$ (we do not assume $div_h u \in L^\infty_{t,x}$). We show that by carefully choosing the noise vector fields $a_i$ (and the number $N$ of them), the initial-value problem is well-posed in the class of weak $L^2$ solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this "regularization by noise" result reveals a link between the nonlinear structure of the underlying domain $M$ and the noise, a link that is somewhat hidden in the Euclidian case ($a_i$ constant) \cite{Beck:2019,Flandoli-Gubinelli-Priola,Neves:2015aa}. The proof is based on an a priori estimate in $L^2$, which is obtained by a duality method, and a weak compactness argument.

math.AP↗

arXiv2024

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the $L^1$ sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the ``zero-filter'' (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the ``filter size'' and (ii) the dissipation of an arbitrary convex entropy.

math.AP↗

arXiv2024

Global existence of dissipative solutions to the Camassa--Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--Itô correction term.

math.AP↗

arXiv2023

Compactness estimates for difference schemes for conservation laws with discontinuous flux

We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t$ is $BV$-regular and may exhibit discontinuities along curves in the $(x,t)$ plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and the existence of one entropy function. While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes. Notably, our compactness estimates are new even in the homogeneous case ($k\equiv 1$).

math.NA↗

arXiv2023

A dynamic capillarity equation with stochastic forcing on manifolds: a singular limit problem

We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold $(M,g)$. \begin{equation*}\tag{P} d \left(u_{\varepsilon,δ}-δΔ u_{\varepsilon,δ}\right) +\operatorname{div} f_{\varepsilon}(x, u_{\varepsilon,δ})\, dt =\varepsilon Δu_{\varepsilon,δ}\, dt Φ(x, u_{\varepsilon,δ})\, dW_t, \end{equation*} where $f_{\varepsilon}$ is a sequence of smooth vector fields converging in $L^p(M\times \Bbb{R})$ ($p>2$) as $\varepsilon\downarrow 0$ towards a vector field $f\in L^p(M;C^1(\Bbb{R}))$, and $W_t$ is a Wiener process defined on a filtered probability space. First, for fixed values of $\varepsilon$ and $δ$, we establish the existence and uniqueness of weak solutions to the Cauchy problem for (P). Assuming that $f$ is non-degenerate and that $\varepsilon$ and $δ$ tend to zero with $δ/\varepsilon^2$ bounded, we show that there exists a subsequence of solutions that strongly converges in $L^1_{ω,t,x}$ to a martingale solution of the following stochastic conservation law with discontinuous flux: $$ d u +\operatorname{div} f(x, u)\,dt=Φ(u)\, dW_t. $$ The proofs make use of Galerkin approximations, kinetic formulations as well as $H$-measures and new velocity averaging results for stochastic continuity equations. The analysis relies in an essential way on the use of a.s.~representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.

math.AP↗